OF ARTS AND SCIENCES. 311 



this series is formed by multiplying the preceding term by a factor of 



the form • — . But n continually increases, while Tq remains constant. 



Hence, however great a finite value T^" may have, n will finally become 

 greater than T^, and still remain finite. It will therefore be possible 

 to take n so great, and still finite, that the ratio between the last two 

 terms of S,^ shall be finitely greater than unity. Insert h, a finite 

 quantity, between this limit and unity. Then 



-> k > 1, whence 



«n + i <p «rt + 2 <-^^ <p' ^^^ 1" general a„+^ <— . Ihen, 

 designating by S^^p the p terms from a„ to a„_|_j,_i inclusive, we have 



Sn,p = «« + a,i + 1 4- . . . -f an J^p - 1 <an (1 + - + ... -|- j^^zzj) 

 or 



<"«{i-:l~ ^i)' . '^"•^ <'''' [k^^i ~ kp(k - 1))- 



When JO = 00, S„^p <a„ , , and therefore S„^„ is finite, and since 



S^ is finite, therefore *S',^ -|- S,,^ ^ =z S is finite, and S is convergent. 

 But T^ — <^ S and . • . T^ is finite. If the tensor of a quaternion 

 is finite, the quaternion is finite ; and hence, as was to be proved, the 

 series 



^=1 +? + ?, + ••• 



is convergent. This series is the exponential of quaternions, and may 

 be designated by the usual symbol 6'- It is a quaternion complanar 

 with q. 



2*^. In ordinary scalar analysis, the fundamental principle of the 

 exponential function consists in the relation 



01 + 9'= 6^6^', 



where q and q' are numbers. In quaternion analysis, this principle 

 holds good only for complanar quaternions, in which the commutative 

 property always obtains. The proof I give is pi-ecisely similar to 

 that given for complex scalar quantities by MM. Briot et Bouquet, 

 Fonc. Ellipt. p. 89. 



